Article
pubs.acs.org/JPCB
Interface-Limited Growth of Heterogeneously Nucleated Ice in
Supercooled Water
Razvan A. Nistor,† Thomas E. Markland,‡ and B. J. Berne*,†
†
Department of Chemistry, Columbia University, 3000 Broadway, MC 3103, New York, New York 10027, United States
Department of Chemistry, Stanford University, 333 Campus Drive, Stanford, California 94305, United States
‡
ABSTRACT: Heterogeneous ice growth exhibits a maximum in freezing
rate arising from the competition between kinetics and the thermodynamic
driving force between the solid and liquid states. Here, we use molecular
dynamics simulations to elucidate the atomistic details of this competition,
focusing on water properties in the interfacial region along the secondary
prismatic direction. The crystal growth velocity is maximized when the
efficiency of converting interfacial water molecules to ice, collectively known
as the attachment kinetics, is greatest. We find water molecules that contact
the intermediate ice layer in concave regions along the atomistically
roughened surface are more likely to freeze directly. An increased
roughening of the solid surface at large undercoolings consequently plays
an important limiting role in the rate of ice growth, as water molecules are
unable to integrate into increasingly deeper surface pockets. These results
provide insight into the molecular mechanisms for self-assembly of solid
phases that are important in many biological and atmospheric processes.
1. INTRODUCTION
Self-assembly of a disordered liquid to an ordered solid is one
of the most basic physical processes that occurs in nature.1,2 Of
these processes, the homogeneous and heterogeneous growth
of ice from liquid water has attracted considerable attention
because of its relevance in atmospheric physics,3−7 cryobiology,8,9 and in the antifreeze and food preservation
industries.10−13 However, while the thermodynamics of
freezing is largely understood,14,15 the molecular details of the
freezing process are less well-established.
Because of the constantly evolving nature of ice growth, it is
difficult to probe the moving solid−liquid interfacial region
experimentally at the microscopic level. Measurements have
shown the ice−water interface to be on the order of ∼1 nm
wide, or three water layers thick, near equilibrium at the
melting temperature.16 Experiments observing dendritic ice
growth have measured maximum growth rates on the order of
10 cm/s for the basal plane at temperatures ΔTM = −18 K
below the melting point.17−19 Experimental deviations of
growth rates from theoretical predictions20 were suggested to
be due to an unaccounted-for competition between collective
molecular attachment (freezing) and detachment (melting)
processes near the solid surface.20−25
Molecular dynamics (MD) simulations have proved a useful
tool for probing more directly the microscopic properties of the
(moving) solid−liquid interface.26−43 These studies have also
shown the interfacial region is about three water layers wide
and consists of a slushy mix of ice and liquid features whose
dynamical properties are greatly arrested compared to the bulk
liquid.30,33 Ice growth rates were shown to reach a maximum
© 2014 American Chemical Society
deep within the supercooled regime, initially increasing as the
temperature was lowered below the melting point but then
decreasing upon further undercooling below a characteristic
temperature.37,42,44 Ideally, one would like to establish how the
structure, shape, dynamics, and molecular attachment rates at
the ice surface change with external conditions near this
crossover point. If the self-assembly mechanism is a
competition between the rates of attachment and detachment
at the liquid−solid contact, what microscopic properties at the
interface favor molecular retention or loss? Furthermore, what
microscopic properties explain why the freezing rate reaches a
maximum in the supercooled regime?
Here, we use molecular dynamics simulations to investigate
how temperature affects the attachment kinetics of water to the
secondary prismatic face of ice Ih. We find the temperature
dependence of the ice growth rate reaches a maximum when
the microscopic efficiency of converting interfacial water to ice
is maximum. This efficiency is limited at higher temperatures
because of repeated melting and surface migration45 events
across the interfacial regions. At lower temperatures, the
interplay between the roughening of the ice surface and
increased tetrahedrality of the liquid42 play important limiting
roles. We find molecules that make contact with the
intermediate ice layer in concave regions are more likely to
freeze directly. Molecules that make contact with regions of
higher curvature tend to escape back into the liquid.
Received: September 3, 2013
Revised: December 30, 2013
Published: January 6, 2014
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was chosen to contact the water region because it is the fastestgrowing face of ice.37 The resulting configuration consisted of a
sheet of ice (S) and bulk liquid (L) separated by two ice−liquid
interfacial regions (I1, I2) shown schematically by the vertical
lines in Figure 1a. Ice growth was monitored perpendicular to
the interface along the Lx direction. Two system sizes were used
to assess finite size effects. Ten trajectories at each temperature
were performed using a small simulation cell containing 2696
water molecules with approximate dimensions 9.1 × 3.1 × 2.9
nm3. Additionally, three trajectories were performed at each
temperature using a large simulation cell, which had nine times
the cross-sectional surface area of the small system and
contained 24 264 water molecules. Data was gathered only
until each trajectory was 60% frozen. This ensured that the
close proximity of the two interfacial regions in the simulation
cell did not affect the analysis at longer times.
Consequently, water molecules are unable to rearrange and fit
into increasingly deeper surface pockets at very low temperatures. Our results highlight the important role played by
interfacial water properties in determining the rate of
heterogeneous ice growth at increasingly larger undercoolings.
2. SIMULATION DETAILS
Molecular dynamics simulations of the TIP4P/2005 water
model46 were performed using the GROMACS47 package on
the ice−liquid system shown in Figure 1a under isobaric−
3. INTERFACE IDENTIFICATION
We developed a robust scheme for classifying the evolving ice,
liquid, and interfacial regions throughout the freezing process.
This classification was accomplished by employing a suitable
order parameter to distinguish between local ice- and liquidlike
structures. Then profile functions of this order parameter were
used to identify the instantaneous interface as described below.
3.1. Instantaneous Molecule Classification. We used a
local tetrahedral order parameter to classify whether the local
structure about a water molecule was ice- or liquidlike53
Sθ =
Figure 1. (a) Initial configuration used in the simulations with
schematic outlines of the two interfacial regions (I1, I2) separating the
solid (S) and liquid (L) phases. Ice grows along the Lx-direction. (b)
Tetrahedral order parameter distributions where angle brackets ⟨S⟩(r,t)
denote both spatial averaging over several configurations of the
trajectory, as well as time-based (exponential) smoothing of the
resultant instantaneous order parameters for each molecule in separate
bulk ice and liquid simulations. Overlap between ice and liquid
distributions is significantly reduced using this scheme. The ice
distribution is normalized by 1/45 to make the y-axis scale more
tractable for viewing.
3
32
3
2
⎛
1⎞
⎜cos θ +
⎟
jk
⎝
3⎠
k=j+1
4
∑ ∑
j=1
(1)
where the summations extended over all hydrogen bond angles
θjk defined by the nearest four neighboring oxygen atoms
around a given molecule. To enhance our ability to distinguish
between ice- and liquidlike distributions (which can overlap up
to 20% at low temperatures), we used a combination of
position averaging and exponential time smoothing of the
trajectories as described in Appendix A. A water molecule was
labeled as icelike when its enhanced order parameter ⟨Sθ⟩(r,t)
was less than a carefully selected threshold criteria Stol. Using
the smoothing procedure, near perfect separation between ice
and liquid configurations can be obtained with less than 4%
overlap between bulk ice and bulk liquid distributions at 220 K,
as shown in Figure 1b. The critical threshold value were chosen
to be Stol = 0.002 for 220 K and Stol = 0.005 for all other
temperatures. At 300 K, no water molecules were misclassified
as icelike in simulations of the bulk liquid. All other observables
were calculated from the raw unaveraged trajectories.
3.2. Instantaneous Interface Classification. The instantaneous positions of the interfacial regions were identified
from profile functions of the tetrahedral order parameter
⟨Sθ⟩(r,t) projected along the direction of ice growth. This
approach is similar to methods used in previous studies to
define the extent of the interfacial region.26,30,54 Here, we track
two interface boundaries as shown in Figure 2a: the solid
interface (SI) and the liquid interface (LI). These dividing
surfaces (dashed vertical lines in the figure) were identified
using the procedure described in Appendix B.
Figure 2b shows the projected density along the scaled
simulation box for a typical trajectory at 220 K obtained using
this scheme. From these boundary lines, the widths of the
interfacial regions (I1, I2) and the thicknesses of the ice (S) and
liquid (L) regions can be monitored as freezing occurs, as
isothermal conditions. Periodic boundary conditions were
employed with the long-range electrostatics treated using
particle-mesh Ewald summation.48 The pressure was kept at 1
bar using an anisotropic Parrinello−Rahman49 barostat with a
time constant of 10 ps. Constant temperature conditions were
imposed using a Langevin thermostat with a time constant of
4.0 ps, which quickly removes latent heat from the system.37,44
Thermostat couplings from 0.1 to 100.0 ps did not affect the
observed freezing rates at 240 K within statistical error.
Water intermolecular interactions were modeled using the
fully atomistic TIP4P/2005 potential. This model was chosen
because it has been shown to accurately reproduce the highdensity phase diagram of water and ice46 as well as the
dynamics of water in the supercooled regime.50 The melting
temperature of the model is 250 K,51 which is considerably
more accurate compared with other simple point charge water
potentials given the other advantages of this model. Freezing
rates obtained with the model are also close to experimentally
observed values.37
The initial ice structure was prepared according to the
Bernal−Fowler rules52 and brought in contact with an
amorphous water configuration approximately three times the
thickness of the ice region. The secondary prismatic plane of ice
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Figure 3. (a) Average crystal growth velocities RG for the secondary
prismatic face of ice Ih for the two system sizes. (b) The retention
probability α defining the system efficiency of converting liquid water
molecules to ice.
model.41 Experimental studies of the growth of ice dendrites
report growth velocities of 10−12 cm/s for the fastest-growing
ice faces at temperatures ΔTM = −18 K below the melting
point.17,55
The maximum in the growth rate in Figure 3a is
characteristic of a crossover from thermodynamically driven
to kinetic-limited crystal growth.1,37,42,44 From 247 to 240 K,
the rate of crystal growth increases by a factor of ∼2 as the
chemical potential difference between the liquid and solid
phases increases (i.e., μL − μS > 0) and ice becomes
thermodynamically more favorable.15 Below 240 K, the growth
rate progressively decreases and at 220 K is a factor of ∼5
smaller than the maximum. This decrease is described as arising
from increasing kinetic barriers governing activated processes
such as diffusion, which become rate-limiting below 240 K.1,37
We note the lowest temperature, 220 K, is below where the
peak in the isobaric heat capacity occurs for this water model
(∼225 K, which signifies the onset of water’s so-called no-man’s
land)56−59 and was included to see if the scaling of the growth
process continues as the system enters this deeply cooled
regime.
To assess the efficiency with which water molecules in the
interfacial regions are incorporated into the ice phase at the
varying temperatures, we consider the interfacial retention
probability α
Figure 2. (a) Tetrahedral order parameter binned along direction of
ice growth and resultant profile function used to identify the solid
interface (SI) and liquid interface (LI) dividing surfaces. (b) Contour
map of the density profile evolving during a typical trajectory with the
solid lines giving the instantaneous interfacial boundaries (SI and LI)
for each interfacial region. (c) Average resultant thicknesses of the ice
and liquid and the widths wreg of the interfacial regions (I1, I2).
shown in Figure 2c. Importantly, the interfacial widths remain
roughly constant throughout the simulation and are on the
order of ∼1 nm, or approximately three water layers, consistent
with experimental measurements16 and other MD studies.30,38,43,54
Because ice growth does not proceed on a layer-by-layer
basis along the prismatic directions,28,33,39 we modeled the
rough SI and LI surfaces by dividing the cross-sectional area of
the simulation box into a 2D array of fibers extending the entire
length of the system along the direction of ice growth. Each
square fiber was defined by a feature size of df = 0.6 nm sides.
The envelope functions, and subsequently the positions of the
SI and LI boundaries, were formed separately in each fiber.
4. RESULTS AND DISCUSSION
4.1. Growth Rate Maximum. The measured ice growth
rates RG along the secondary prismatic direction for different
temperatures are shown in Figure 3a. The growth rate profile
reaches a maximum of 9.6 ± 0.5 cm/s at 240 K (ΔTM = −10
K), in good agreement with previous results using this water
α=
ΦF
ΦF + ΦE
(2)
where ΦF is the flux (number of molecules per square
nanometer per nanosecond) of liquid molecules that
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irreversibly freeze to the solid surface and ΦE is the flux of
liquid molecules that enter the interfacial region but later
escape to the liquid without freezing. Written in this way, α is
the liquid-to-ice conversion efficiency of the system, or the
percentage of water molecules that freeze from the total
number of water molecules that cross into the interfacial
regions (I1 and I2) from the liquid.
The temperature dependence of the retention probability α
in Figure 3b follows the same trend observed in the growth rate
profile, exhibiting a maximum at 240 K and minima at the
lowest and highest temperatures 220 and 247 K for both system
sizes studied. This temperature dependence is intuitively
expected because the crystal growth velocity is microscopically
determined by the rates which molecules become incorporated
into the solid surface. Crystal growth will be limited if the
conversion of liquid molecules to ice is low, as is the case at 220
and 247 K in Figure 3b.
The individual contributions to the retention probability in
Figure 4 allow the origins of the freezing efficiency to be
assessed. As can be seen from Figure 4a, the flux of molecules
that irreversibly freeze ΦF increases by nearly a factor of 2 as
the temperature is lowered from 247−240 K and ice becomes
thermodynamically more favorable. There are fewer irreversible
freeze events at 247 K because of an increased propensity to
melt, or to detach from the icelike layers near the solid surface
(see next paragraph), which is intuitively expected near the
melting point. Below 240 K, however, ΦF begins to decrease
significantly from its maximum value, by up to a factor of ∼4 at
220 K (a similar change in freezing rate is observed in Figure
3a). While the increased tetrahedrality of the liquid42 and
corresponding slow-down in water dynamics44 are expected to
contribute to the observed decrease in freeze events, we
examine further how changes in the interfacial structure can
also play a limiting role in the attachment kinetics at low
temperatures.
The escape flux of molecules that enter the interfacial region
but return to the liquid is shown in Figure 4b. The
nonmonotonic behavior shows that the number of escape
events is (somewhat unintuitively) greater at both 220 and 247
K. An increased propensity to detach from the ice surface (or
icelike layers near the ice surface) plays an increasingly greater
role in the escape flux as the temperature increases. Figure 4c
shows how the flux ΦM of frozen molecules that detach from
the ice and icelike planes near the solid surface, i.e., melt events,
increases with temperature and proximity to the liquid region
(dashed lines), as is intuitively expected. At temperatures below
240 K, however, increasing values for ΦE are more puzzling.
This greater propensity to return to the liquid will be attributed
to a roughening of the intermediate ice layers, as will be
discussed below.
The microscopic population analysis shows that although the
system has similarly low retention probability and ice growth
rates at 220 and 247 K, the reasons for the limited growth
velocities are quite different. At high temperatures, the growth
rate is limited because of an increased melting propensity. At
low temperature, the growth rate is limited by a sharp decrease
in the number of direct freezing events because of an inability
to convert interfacial water molecules into ice before these
molecules also escape back to the liquid. In the following
sections, we analyze how the microscopic dynamics and
interface topology contribute to these observations.
4.2. Effects of Interface Topology. It is constructive to
analyze the collective freezing and melting events at the solid
Figure 4. (a) Flux ΦF of molecules that irreversibly freeze to the solid
phase. (b) Flux ΦE of molecules that cross into the interfacial regions
but later escape to the liquid without contacting the ice surface. (c)
Flux ΦM of molecules that escape to the liquid (melt) from the ice
surface (solid lines) and icelike planes progressively farther into the
interfacial region (dashed lines).
interface (SI) guided by a simple model. As discussed in
Instantaneous Interface Classification, we approximate the
roughening of the interfacial dividing surfaces by fitting
envelope functions of ⟨Sθ⟩(r,t) in a series of discretized
rectangular fibers extending the length of the simulation box.
If the fluctuating position of the SI in each fiber is treated as a
biased random walker, where a step forward represents a
freezing event and a step backward represents a melting event,
and where the random walk is biased by the degree of
undercooling, then ensemble averages over all the fibers derived
using the moment-generating function for continuous walks
will satisfy60
⟨δx(τ )⟩ = ⟨xi(t0 + τ ) − xi(t 0)⟩ = (k f − k b)aτ
(3)
and
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⟨δx 2(τ )⟩ = ⟨δx(τ )2 ⟩ − ⟨δx(τ )⟩2 = (k f + k b)a 2τ
Article
thicker fibers up to df = 1.0 nm, (2) interfacial profile functions
constructed from the total density ρ,30 and (3) trajectories
generated using the standard TIP4P water model61 (at relative
undercoolings).
The forward rate kf increases by a factor of ∼1.3 from 247−
240 K, and progressively decreases by a factor of ∼1.5 from
240−220 K. The backward rate shows a stronger nonmonotonic temperature dependence: decreasing by a factor
of ∼5 from 247 K to the minimum at 240 K and increasing by
roughly the same factor upon further cooling to 220 K. The
nonmonotonic behavior in Figure 5a shows that the difference
between forward and backward processes, and consequently the
growth rate RG = (kf − kb)a, is greatest at 240 K and lowest at
both 220 and 247 K. While low thermodynamic driving force
and increased melting propensity can intuitively describe the
small difference in (kf − kb)a at 247 K (see also Figure 4a,b), it
is instructive to analyze how changes in interfacial structure
coupled with arrested water dynamics44 and increased
tetrahedrality of the liquid42 can limit the attachment kinetics
that microscopically underpin the forward and backward rates
at low temperatures.
The microscopic features of the interfacial regions are
intricately linked to nonmonotonic temperature dependence of
the forward and backward processes at increasingly lower
temperatures. Notably, the structural characteristics of the
solid−liquid interface scale differently than in solid−vapor
systems. Figure 5b shows the root-mean-squared (RMS)
deviation of the discretized fibers used to define the
instantaneously roughened solid interface for all temperatures.
The figure shows that the dividing surface becomes rougher as
the temperature is lowered, consistent with previous simulations,38 but opposite to what is observed at the ice−vapor
interface.14,62 The RMS deviation is slightly larger than 0.3 nm
at the coldest temperatures, corresponding approximately to an
extra ice layer, or half a hexagon as viewed from the basal
direction. The larger systems show higher RMS deviations at
higher temperatures, which is a notable finite size effect.
Small systems in particular can lead to overestimated growth
kinetics.37 To understand how this arises, Figure 5c shows the
roughness correlation length ξ extracted from the height
difference spatial correlation function for the large systems63,64
(4)
where xi(τ) is the position of the ith cell or fiber at time τ; kfa
and kba are the forward (freezing) and backward (melting)
rates, respectively; and a is the average measured step size.
Equation 3 gives the growth rate (or velocity) as RG = (kf − kb)
a. Equation 4 is a measure of the spread of the random walker
trajectories. Using these relations, the forward and backward
rates can be extracted for each temperature.
Figure 5a shows the nonmonotonic temperature dependence
of the forward (kf) and backward (kb) rates for the two system
sizes using a feature size of df = 0.6 nm for the square fibers
defining the discretized random walkers. We have separately
verified that the trends are qualitatively reproduced using (1)
Ch(ΔR) = ⟨|h(r) − h(r′)|2 ⟩
(5)
where h(r) is the height of the discretized fiber (in this case, its
extent along the x-direction) at position r along the crosssectional area of the roughened surface and ΔR = r − r′. The
roughness correlation length was extracted by fitting the
function Ch(ΔR) to the form64
Ch(ΔR) = 2σ 2{1 − exp[− (ΔR /ξ)2P ]}
(6)
where σ is the standard deviation in heights, ξ the correlation
length, and P a roughness exponent, typically between 0.5 and
0.6 for the large systems. The temperature dependence of ξ can
be fitted to the Kosterlitz−Thouless scaling relation (dashed
line), which increases exponentially and diverges at the
roughening transition temperature near the melting point.65
At 247 K, the roughness correlation length is nearly 1.6 nm,
almost half the size of the large system simulation cell. These
data indicate that finite size effects will become prevalent if
long-wavelength capillary waves driving the roughening
transition are damped out because of small simulation cells.
Higher observed growth rates in sufficiently small systems
Figure 5. (a) Forward (kf) and backward (kb) rates representing
freezing and melting processes of the solid interface for the two
different sized systems. Results are multiplied by the measured average
step size a = 0.23 nm, which roughly equals the spacing between
successive ice planes. (b) Root-mean-squared deviation of the
discretized roughened ice surface σRMS. (c) Spatial correlation lengths
ξ for the large systems extracted by fitting the height difference
correlation function shown in the inset as an example at T = 230 K.
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could result because of the appearance of defected surface
motifs that aid molecular rearrangement near the solid
surface.63,66
The increase in the roughness correlation length with
temperature in Figure 5c shows structural features are more
correlated at higher temperatures. Consequently, the solid
interface appears smoother, with a smaller RMS deviation in
the height profiles of the discretized fibers. At lower
temperatures, structural correlations occur over shorter length
scales and the ice surface becomes rougher.
To gain insight into which properties of the roughened
interface inhibit molecular retention at low temperatures, we
track where incoming molecules contact the intermediate ice
layer (IIL), which is typically composed of a few roughened
water layers above the ice boundary as described in Appendix
A. We divide the flux of these incoming molecules into
molecules that freeze and molecules that escape back to the
liquid without being incorporated in the solid phase. We
measure the average distance d of a water molecule to this IIL
and the curvature κ of the roughened surface at the contact
point. Contact is defined by the formation of hydrogen bonds
between the incoming water molecule and molecules that are
part of the IIL. The mean curvature κ at each point of the
atomistically rough intermediate ice surface was calculated from
the average of the principal curvatures, κ = 1/2(κmin + κmax),
where κmin and κmax are the eigenvalues of the shape operator
representing the minimum and maximum degree of deflection
of a surface at a given point.67
As an example, Figure 6a shows the curvature distributions
for the freezing and escaping populations averaged over all
trajectories at 230 K. Molecules that directly freeze tend to
dock in concave valleys, or regions of negative (or near zero)
mean curvature on the roughened intermediate ice surface.
Conversely, molecules that escape back into the liquid tend to
bind to peaks, or regions of high mean curvature farther away
from the solid interface. Particles subsequently have a greater
propensity to escape from convex surfaces than from concave
surfaces of ice, as seems to be the case for liquid−vapor
interfaces as well (see refs 63 and 68.).
Figure 6b measures the degree of overlap between the freeze
and escape distributions for both observables, quantified by
taking differences in the mean values: Δκ = ⟨κescape⟩ − ⟨κfreeze⟩
and Δd = ⟨descape⟩ − ⟨dfreeze⟩. The differences in both Δκ and
Δd decrease as the temperature approaches the melting point,
where the solid surface is much flatter. Analogously, there are
fewer peaks and valleys for the smoother interfaces at higher
temperatures and less of a docking preference between freeze
and escape populations. At large undercoolings, however, the
separation between distributions is larger, indicating that escape
events preferentially bind to regions of higher curvature. These
roughened structural features may develop in order to expose
the more stable prismatic face to the liquid contact.32,69
However, surface roughening has been noted to appear on the
prismatic and basal faces of ice as well.39
To asses how the roughened structural features of the ice
surface impact the growth kinetics, Figure 6c shows Pearson’s
statistical correlation coefficient gcorr between the instantaneous
growth rates RG and surface roughness σRMS for the large
systems. Instantaneous growth rates were obtained from 2.0 ns
moving windows centered at each frame of the trajectories. The
coefficient is −1.0 when growth rates are completely
anticorrelated with surface roughness. When gcorr is 0.0, the
two measures are uncorrelated. As can be seen in the figure,
Figure 6. (a) Mean curvature distributions κ of freeze and escape
populations at T = 230 K. The inset shows a colormap of the resulting
curvature values atop an instantaneous configuration of the
intermediate ice layer. (b) Differences between freeze and escape
distributions for the two measures, ⟨Δd⟩ and ⟨Δκ⟩, for the two system
sizes. (c) Statistical correlation gcorr between low growth rate and high
surface roughness for the large systems.
gcorr approaches 0.0 at high temperatures and −0.51 ± 0.08 as
the temperature is lowered to 220 K. Consequently, low growth
rates are increasingly associated with high surface roughness
below the temperature of maximum crystallization.
5. SUMMARY AND CONCLUSIONS
Using molecular dynamics simulations, we have identified
structural features of the ice−liquid interface along the
secondary prismatic direction that affect crystal growth
velocities in the supercooled regime. Near the melting point,
the freezing rate is limited by surface depletion events, as
molecules detach from the ice and migrate back to the liquid.
At much lower temperatures, the topology of the roughened
intermediate ice layer plays an important limiting role hindering
ice growth. The decrease in the interfacial retention probability
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locations of the SI and LI dividing surfaces as shown by the
dashed vertical lines in Figure 2a.
The distributions were accrued over 200 ps time windows.
This sampling corresponded to roughly one-tenth the time it
took the solid interface to pass through the next layer of ice in
the fastest-growing trajectories. The positions of the SI
boundaries were aligned with the nearest ice plane at each
time step because the lower shoulders of the fitted profile
function were not necessarily concomitant with the outermost
ice layer. Once the interfacial boundaries were established,
molecules could be identified as being in the solid (S), liquid
(L), or in one of the two interfacial (I1, I2) regions at any given
time in the trajectory. To remove rapid recrossing events across
the boundaries, a molecule was required to reside a minimum
of 200 ps in a region before it counted towards the population
statistics. Molecular configurations and surfaces were rendered
using MATLAB and the Visual Molecular Dynamics (VMD)
package.70,71
at low temperatures is due to the appearance of high-curvature
structural motifs. Along with the increased tetrahedrality of the
liquid,42 these roughened structural profiles limit the crystal
growth velocities at larger undercoolings, as the liquid is unable
to adjust to the required surface geometry.
At the temperature of maximum crystallization, the efficiency
of converting interfacial water to ice is maximized. The rates
between competing attachment and detachment reactions in
the interfacial region is greatest at this temperature. Notably,
molecular detachment rates leading to surface melting are
minimized when the crystallization rate is maximized. The
liquid is best able to adjust and fill surface pockets at the
temperature of maximum growth.
These insights into the molecular scale rate-limiting
processes for heterogeneous ice nucleation should prove useful
in analyzing how other perturbations, such as the presence of
solutes, affect the interfacial region and freezing rates. Such an
understanding is vital for unraveling the ice growth inhibition
mechanisms of antifreeze proteins in biological systems and for
industrial cryogenics applications.
■
■
AUTHOR INFORMATION
Corresponding Author
APPENDIX
*E-mail: bb8@columbia.edu.
A. Ice−Water Selectivity
Notes
In order to enhance the selectivity between ice- and liquidlike
local configurations using the tetrahedral order parameter in eq
1, the atomic positions of the trajectories were first averaged
over 5 ps windows to reduce thermal and librational noise in
the oxygen atom positions. This time is shorter than the ∼10 ps
characteristic water reorientation time for this water model at
250 K,50 which is a higher than any temperature used here and
so is much shorter than the characteristic time in which a
molecule can interconvert between ice and water in our
trajectories. These averaged positions were then used to
evaluate the tetrahedral order parameter using eq 1 for each
water molecule.
The order parameter history of each water molecule from the
position-averaged trajectories was then exponentially timesmoothed using ⟨Sθ⟩(r,t) = α⟨Sθ⟩(r) + (1 − α)⟨Sθ⟩(r,t−1), where
⟨Sθ⟩r is the instantaneous position-averaged order parameter of
a given molecule at time t, ⟨Sθ⟩(r,t−1) the smoothed order
parameter of the previous time step t − 1, and α the smoothing
parameter. We found α ∼ 0.3 gave adequate separation
between bulk ice and liquid distributions at low temperatures.
This time-based smoothing was used to inhibit instantaneous
tetrahedral configurations from contributing to the analysis,
which may spontaneously occur even at high temperatures.
Molecules within the interfacial regions whose order
parameter ⟨Sθ⟩(r,t) was greater than the ice threshold criteria
Stol but less than 75% of the liquid value at the given
temperature were labeled as intermediate ice.42 A molecule
retained its ice, liquid, or intermediate ice label for the duration
of the 5 ps position-averaging window. These labels were used
only for population analysis.
The authors declare no competing financial interest.
■
ACKNOWLEDGMENTS
The authors thank Valeria Molinero for her insightful
comments on an early version of this manuscript. This research
was supported by a grant to B.J.B. from the National Science
Foundation (NSF-CHE-0910943). We thank the Computational Center for Nanotechnology Innovations at Rensselaer
Polytechnic Institute and the Extreme Science and Engineering
Discovery Environment resources at the Texas Advanced
Computing Center for providing computational facilities to
support this project.
■
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